Lesson 2

Linear equations/inequalities; simultaneous equations

<p>Learn about Linear equations/inequalities; simultaneous equations in this comprehensive lesson.</p>

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Why This Matters

Imagine you're trying to figure out how many candies you can buy with your pocket money, or how to share pizza fairly with your friends. That's where linear equations and inequalities come in! They're like secret codes that help us solve puzzles about numbers and amounts in everyday life. Sometimes, you have more than one puzzle to solve at the same time. Maybe you need to know how many apples AND how many bananas you can buy, but you have a total budget and a total number of fruits you want. That's when **simultaneous equations** become your superhero tool, helping you find the answers to multiple questions all at once. Learning these topics isn't just for school; it helps you think logically and solve problems, whether you're planning a party budget or figuring out the best deal at the shop!

Key Words to Know

Equation — A mathematical statement showing two expressions are equal, like a balanced seesaw.

Variable — A letter (like x or y) that represents an unknown number we want to find.

Linear Equation — An equation where the variable's highest power is 1, meaning it will graph as a straight line.

Inequality — A mathematical statement showing that two expressions are not equal, using symbols like <, >, ≤, or ≥.

Simultaneous Equations — A set of two or more equations with the same variables that must be solved together to find values that satisfy all of them.

Coefficient — The number multiplied by a variable in an algebraic term (e.g., in '3x', '3' is the coefficient).

Substitution Method — A technique to solve simultaneous equations by solving one equation for a variable and plugging that expression into the other equation.

Elimination Method — A technique to solve simultaneous equations by adding or subtracting the equations to make one variable disappear.

Solution — The value(s) of the variable(s) that make an equation or inequality true.

Isolate the Variable — The process of rearranging an equation or inequality to get the variable by itself on one side.

What Is This? (The Simple Version)

Think of an equation like a balanced seesaw. On one side, you have some numbers and letters (called variables, which are like mystery numbers waiting to be discovered). On the other side, you have more numbers. The equals sign (=) in the middle means both sides must weigh exactly the same.

A linear equation is a special kind of seesaw where the mystery numbers (variables) are just plain old numbers, not squared or cubed. So, you might see 'x + 5 = 10', but not 'x² + 5 = 10'. It makes a straight line if you draw it on a graph.
An inequality is like a seesaw that isn't perfectly balanced. Instead of an equals sign, it uses symbols like < (less than), > (greater than), ≤ (less than or equal to), or ≥ (greater than or equal to). For example, 'x < 5' means 'x' can be any number smaller than 5, like 4, 3, 2, or even 4.999!
Simultaneous equations are like having two or more balanced seesaws that are connected. The mystery numbers (variables) on one seesaw are the same mystery numbers on the other seesaw. You need to find the values for these mystery numbers that make all the seesaws balance at the same time.

Real-World Example

Let's say you and your friend are buying snacks. You know that a packet of crisps (let's call its price 'c') and a chocolate bar (let's call its price 'h') together cost $3. This is our first equation: c + h = 3.

Then, your friend tells you that two packets of crisps and one chocolate bar cost $5. This is our second equation: 2c + h = 5.

Now, we have two equations that are true at the same time, and we want to find out the individual price of a packet of crisps ('c') and a chocolate bar ('h'). This is a perfect job for simultaneous equations!

From the first equation (c + h = 3), we know that h = 3 - c.
We can now 'substitute' this into the second equation. Instead of 'h', we write '3 - c': 2c + (3 - c) = 5.
Simplify: 2c - c + 3 = 5, which means c + 3 = 5.
Subtract 3 from both sides: c = 2. So, a packet of crisps costs $2.
Now that we know c = 2, we can put it back into our first equation: 2 + h = 3.
Subtract 2 from both sides: h = 1. So, a chocolate bar costs $1.

See? We solved the mystery prices using simultaneous equations!

Solving Linear Equations (Step by Step)

Solving a linear equation means finding the value of the mystery number (variable) that makes the seesaw balance. Our goal is to get the variable all by itself on one side of the equals sign.

Isolate the variable: Imagine the variable is a treasure you want to dig up. You need to remove everything around it.
Do the opposite: If a number is added to the variable, subtract it from both sides. If it's multiplied, divide both sides.
Keep the seesaw balanced: Whatever you do to one side of the equation, you MUST do to the other side to keep it equal.
Example: Solve 3x + 7 = 19.
Subtract 7 from both sides: 3x + 7 - 7 = 19 - 7, which gives 3x = 12.
Divide both sides by 3: 3x / 3 = 12 / 3, which gives x = 4. You found the treasure!

Solving Linear Inequalities (Step by Step)

Solving inequalities is very similar to solving equations, but with one super important rule to remember!

Treat i...

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Solving Simultaneous Equations (Step by Step)

We have two main methods, like two different tools in your toolbox: Substitution and Elimination.

Method 1: S...

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Common Mistakes (And How to Avoid Them)

Here are some common traps students fall into, and how to dodge them!

Forgetting to do the same thing to both sid...

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Exam Tips

1.Always show your working clearly, even for simple steps; marks are often given for method.
2.For inequalities, remember to reverse the sign if you multiply or divide by a negative number.
3.After solving simultaneous equations, substitute your answers back into *both* original equations to check if they are correct.
4.Practice both substitution and elimination methods for simultaneous equations so you can choose the most efficient one for each problem.
5.Read the question carefully to see if it asks for integer solutions (whole numbers) or if decimal/fractional answers are acceptable.